a _- _ _ __ II!!3 4_ I September 1995 ELSEVIER zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA OPTICS COMMUNICATIONS Optics Communications119 ( 1995) 279-282 Optical synthesis of self-Fourier functions Debesh Choudhury a, zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA P.N. Puntambekar a, AK. Chakraborty b a zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA Departmenr of Physics, Indian Institute of Technology, Bombay Powai, Bombay 400 076, India b Department of Applied Physics, University College of Technology, Calcutta UniversityV 92, Acharya Prafulla Char&a Road, Calcutta 700 009, India Received 16 March 1995 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONML Abstract An optical configuration for the realization of the self-Fourier (transform) function of any arbitrary transformable function is proposed. The optical system is a Sagnac-type interferometer coupled with a modified Mach-Zehnder interferometer. The first interferometer displays the object function and its inverted version and the second interferometer simultaneouslyFourier transforms and images them in the same channel thereby resulting in a self-Fourier function. 1. Introduction Self-Fourier (transform) functions (SFFs) are a special class of functions which are their own Fourier transforms. There are enough exercises and practi- cal examples on this subject in standard texts [ l-41. Gaussian and Dirac comb functions are two common examples of SFFs. The pioneering work in this field was done by the famous mathematician G. H. Hardy and his colleague E. C. Titchmarsh, who published this concept sixty four years ago under the title self reciprocal functions [ 51. They showed how to con- struct a self+eciprocal function from any arbitrary transformabl& function of certain class. They further discussed the possibility of getting the complete set of self-reciprpcal functions by solving some integral equations. They also considered the necessary and suf- ficient conditions for a function to be self-reciprocal in cosine (and sine) and Hankel transforms in great details. A considerable interest has been shown in this field of research [ 6-131 in recent years. Caola [ 61 has shown from theoretical considerations how to con- struct a SFF of any arbitrary transformable function. Lohmann et al. [ 71 have generalized it for some other cyclic transforms such as Hartley transform and frac- tional Talbot transform. They have also suggested an optical cyclic transform with odd cycles [ 81. Cincotti et al. [ 91 have given a more generalized definition of SFFs and have shown that any Fourier transformable function is a linear combination of four SFFs whose explicit form can be found. A recent comment by Lip- son [ 101 has stipulated the relation between SFFs and confocal optical resonators. He has also pointed out that SFFs are nothing but the eigen functions of the Fourier transform operator [ lo]. Some more articles have also appeared on SFF, fractional-SFF and related topics [ 1 l-131. But nobody, so far, has considered the problem of the optical synthesis of a SF’F. In this communication we propose a simple technique for the optical realiza- tion of a SFF of any arbitrary transformable function utilizing vector-wave interferometry in a Sagnac-type interferometer (STY) [ 14-161 coupled with a modi- fied Mach;Zehnder interferometer. 0030-4018/95/$U9.50@ 1995 Elsevier Science B.V. All rights reserved SSDIOO30-4018(95)00373-8